f(x) and f′(x) are continuous, differentiable functions that satisfy
f(x)=x^3+4x^2+∫ (from 0 to x)(x−t)f′(t) dt.
What is f′(5)−f(5)?
To find the value of f′(5)−f(5), we first need to find the derivative of f(x) using the given equation. Let's go step by step:
1. We are given the equation:
f(x) = x^3 + 4x^2 + ∫ (from 0 to x) (x−t)f′(t) dt
2. Differentiate both sides of the equation with respect to x.
d/dx[f(x)] = d/dx[x^3 + 4x^2 + ∫ (from 0 to x) (x−t)f′(t) dt]
Using the Fundamental Theorem of Calculus, we know that the derivative of an integral with respect to its upper limit is the original function evaluated at the upper limit. Applying this, the equation becomes:
f′(x) = 3x^2 + 8x + (x - x)f′(x)
3. Simplify the equation:
f′(x) = 3x^2 + 8x + 0
Now we have the expression for f′(x).
4. To find f′(5), substitute x = 5 into the equation:
f′(5) = 3(5)^2 + 8(5) = 75 + 40 = 115
5. Similarly, find f(5) by substituting x = 5 into the original equation:
f(5) = 5^3 + 4(5)^2 + ∫ (from 0 to 5) (5−t)f′(t) dt
Solving this equation will give you the value of f(5).
6. Finally, calculate f′(5)−f(5) by subtracting the values obtained in steps 4 and 5:
f′(5)−f(5) = 115 - f(5)
Note: Since we don't have an explicit expression for f(x), we cannot determine f(5) with the given information. Further calculations or more information about f(x) would be needed to find its specific value at x=5.